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Algorithmic Construction of Gapped Boundaries for Topological Stabilizer Codes: A Route to Low-Overhead Planar qLDPC Codes

Zijian Liang (梁子健)1,†, Bowen Yang (杨博闻)2,†, Joseph T. Iosue3, and Yu-An Chen (陳昱安)1,*

  • *Contact author: yuanchen@pku.edu.cn
  • †These authors contributed equally to this work.
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PRX Quantum 7, 043016 – Published 23 October, 2026

DOI: https://doi.org/10.1103/989h-xjqv

Abstract

Quantum low-density parity-check (qLDPC) codes promise low-overhead fault-tolerant quantum memory, but many leading constructions are naturally defined on periodic lattices, whereas scalable quantum hardware favors planar architectures with open boundaries. In this work, we develop a general operator-algebraic framework and computational algorithms for constructing all gapped boundaries and defects of two-dimensional topological Pauli stabilizer codes. Starting from a truncated bulk code, our method identifies boundary gauge operators, boundary anyons, and their string operators, and then constructs microscopic boundary Hamiltonians through anyon condensation and topological-order completion, thereby eliminating local logical operators near the boundary. We establish a one-to-one bulk-boundary correspondence between bulk anyons and excitations of the anomalous boundary subsystem code, enabling fusion rules, topological spins, and braiding statistics to be extracted directly from the boundary operator algebra. For translation-invariant codes, Laurent-polynomial representations reduce these tasks to finite matrix computations based on Hermite and Smith normal forms, enabling automated constructions for Zd qudits of both prime and composite dimension. We demonstrate the framework on toric, color, double-semion, six-semion, and anomalous three-fermion codes, and apply it to bivariate bicycle codes, revealing large logical dimensions and unusually long translation periods for anyon strings. By systematically converting bulk topological data into explicit boundary and defect Hamiltonians, our framework can serve as a foundation for constructing low-overhead planar qLDPC codes with open boundaries and for engineering fault-tolerant logical operations.

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Many promising quantum error-correcting codes are designed on lattices with periodic boundary conditions, whereas practical devices often favor flat layouts with open boundaries. In this work, we develop a general algorithm for constructing boundaries of two-dimensional topological Pauli stabilizer codes. Starting from the local stabilizers that define the bulk code, our algorithm identifies boundary excitations and their string operators, and constructs gapped boundaries by determining which groups of anyons can condense at the edge. A key insight is that the boundary retains the essential topological information of the bulk, including anyon fusion, topological spin, and braiding. We demonstrate the algorithm across several code families, including bivariate bicycle codes. Our results turn boundary design from a model-specific task into a systematic computation, providing a general route for adapting promising quantum codes to planar architectures and for implementing logical operations through boundaries and defects.

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PRX Quantum 7, 043016– Published 23 October, 2026
  • Received 10 July 2026
  • Accepted 11 September 2026
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